Linear Momentum

and Impulse

Linear Momentum

Linear momentum (p)

is a vector

p is parallel to v

Unit: kg·m/s

The net momentum of

a collection of objects

is the vector sum of

the momentum of

each object

vmp

...

...

332211

321

vmvmvm

pppp

Momentum problem

A bullet of mass 0.005 kg moving at a speed of 100

m/s lodges within a 1-kg block of wood resting

on a frictionless surface and attached to a

horizontal spring of k=50N/m.

a) What is the velocity

of the block the

instant after the

bullet strikes it?

sm

kgkg

smkg

mm

vmv

vmmvm

vmvmvmvm

/50.0

00.1005.0

)/100)(005.0(

)(0

21

11'

'

2111

'

22

'

112211

Momentum problem

A bullet of mass 0.005 kg moving at a speed of

100 m/s lodges within a 1-kg block of wood

resting on a frictionless surface and attached

to a horizontal spring of k=50N/m.

b) What is the maximum

compression of the

spring?

m

mN

smkg

k

mvx

kxmv

EE TT

07.0

/50

)/5.0)(005.1( 2

2

2

212

21

Force and momentum

The time rate of change

of the momentum of a

particle is equal to the

net force acting on

the particle.

dt

pdFnet

“Let the time rate of change of

momentum be with you.”

Force and momentum

In general, force and momentum are related as:

When the mass is constant, this becomes the familiar equation:

dt

dmv

dt

vdmvm

dt

d

dt

pdFnet

)(

amdt

vdmFnet

Non-constant mass

Mass changes for rockets, bags of sand

(with holes in them), and other special

cases.

See below for a changing mass:

http://eepybird.com/dcm1.html

Changing mass problem

A rocket whose initial mass is 850kg consumes

fuel at the rate of 2.3kg/s. The speed of the

exhaust gases relative to the rocket engine is

2800m/s.

What thrust does the

rocket engine provide?

N

skgsm

dt

dmv

dt

dmv

dt

vdm

dt

pdFnet

6440

)/3.2)(/2800(

0

Impulse

Impulse (J) is a change in momentum.

Start with Newton’s second Law:

Which can be written as:

Integrate with respect to time

to find the change in

momentum:

dt

pdF

dttFpd )(

f

i

f

i

t

t

p

p

dttFpdJ )(

Impulse

A 2.0 kg toy car travels at 0.50m/s East before a

turn and at 0.40m/s South after the turn.

What is the impulse (change in momentum) of the

car due to the turn?

smkgji

ismjsmkg

vvm

vmvm

pppJ

if

if

if

/)ˆ8.0ˆ0.1(

ˆ)/50.0(ˆ)/40.0()0.2(

)(

Conservation of Linear

Momentum

The linear momentum (p) of a system is

conserved (does not change) unless the

system experiences an external force.

Problem An 80-kg lumberjack stands at one end of a

floating 400-kg log that is at rest relative to the

shore of a lake.

If the lumberjack jogs to the other end of the log at

2m/s relative to the shore, what happens to the

log while he is moving?

Elastic collision

In an elastic collision objects typically bounce, and

NO energy is lost

Momentum is conserved

Kinetic energy is conserved

afterbefore pp

afterbefore KK

Inelastic collision

In a totally inelastic collision, objects stick together.

Momentum is conserved

Energy is lost to sound, sparks, mechanical

deformation, etc.

afterbefore pp

afterbefore KK

Example of Inelastic collision:

Would you be safer in an old (heavy) 1959

Chevrolet Bel Air or in a newer (lighter)

2009 Chevy Malibu?

http://www.youtube.com/watch?v=fPF4fBGNK0U

Collision review

Momentum is conserved in all collisions

Elastic collisions: no deformation occurs

Kinetic energy is also conserved

Inelastic collisions: deformation occurs

Kinetic energy is “lost”

Perfectly inelastic collisions

Objects stick together; kinetic energy is “lost”

Explosions

Reverse of perfectly inelastic collisions; kinetic energy is “gained”

Collision problem

Two skaters collide and embrace, in a completely

inelastic collision. Dean, of mass 83 kg, is

initially moving east with speed 6.2 m/s. Torvill,

of mass 55 kg, is initially traveling north with

speed 7.8 m/s.

What are the skaters’ speed and direction after the

collision?

408.39

/9.4/86.4

smsmv